Quite expensive at 4 USD but I think it's worth the money - if you have root that is.
You can find files, pack/unpack archives, change permissions, move stuff around, handle multiple tabs and work with SMB and FTP shares, Google Drive, Dropbox.
There might be some free alternatives (ES File Explorer, Root Browser) around but at that time (~2012) there weren't many tools like this..

Another paid tool, even more expensive (6 USD, I might've paid less) but I've used it hundreds of times. It does require root as well.
You can back up an almost complete state of your phone/tablet and restore it to another device.
The backups can be shared between accounts, so you can restore a purchased program with complete settings on another account/device.
I run a weekly backup and just dump the complete contents of the internal and external sd cards to the NAS. Later in time I delete obsolete backups, but they are really handy when things go wrong or if/when you lose your phone.

See in one glance whether something is eating up the CPU. Then open up the developer tools, check 'Show CPU Usage' and see who is at the top. Or open terminal emulator and start killing stuff. Has saved my battery many times in spite of hanging applications.

You've spent days setting up your desktop icons and widgets - then you install an update or switch to a another phone and everything is gone, because the new phone uses a different launcher. This is the solution to that and provides other nice features.